Feynman-Kac formulae

side order of Fokker-Planck

January 27, 2021 — January 27, 2021

Bayes

Monte Carlo

probabilistic algorithms

probability

state space models

statistics

swarm

time series

There is a mathematically rich theory about sequential Monte Carlo filters, and the central tool to make that go seems to be Feynman-Kac formulae.

Related, apparently: Backward SDEs and the Fokker-Planck equation.

1 Use in SMC

In my field, when we see Feynman-Kac the default assumption is that it is providing a central limit theorem for sequential Monte Carlo. The notoriously abstruse Del Moral (2004) and Doucet, Freitas, and Gordon (2001) are commonly regarded as the canonical introductions to that usage. I will get around to understanding them myself eventually, maybe?

Cheng and Reich (2014) translates the Del Moral (French probabilist?) terminology into my more workaday statistician’s language.

2 References

Beck, E, and Jentzen. 2019. “Machine Learning Approximation Algorithms for High-Dimensional Fully Nonlinear Partial Differential Equations and Second-Order Backward Stochastic Differential Equations.” Journal of Nonlinear Science.

Cappé, Godsill, and Moulines. 2007. “An Overview of Existing Methods and Recent Advances in Sequential Monte Carlo.” Proceedings of the IEEE.

Cérou, Moral, Furon, et al. 2011. “Sequential Monte Carlo for Rare Event Estimation.” Statistics and Computing.

Chan-Wai-Nam, Mikael, and Warin. 2019. “Machine Learning for Semi Linear PDEs.” Journal of Scientific Computing.

Cheng, and Reich. 2014. “A McKean Optimal Transportation Perspective on Feynman-Kac Formulae with Application to Data Assimilation.”

Chopin, and Papaspiliopoulos. 2020. An Introduction to Sequential Monte Carlo. Springer Series in Statistics.

Chuang. 2010. “The Feynman-Kac Formula: Relationships Between Stochastic Differential Equations and Partial Differential Equations.”

Del Moral. 2004. Feynman-Kac Formulae: Genealogical and Interacting Particle Systems with Applications.

Del Moral, and Doucet. 2010. “Interacting Markov Chain Monte Carlo Methods for Solving Nonlinear Measure-Valued Equations.” The Annals of Applied Probability.

Del Moral, Hu, and Wu. 2011. On the Concentration Properties of Interacting Particle Processes.

Del Moral, and Miclo. 2000. “Branching and Interacting Particle Systems Approximations of Feynman-Kac Formulae with Applications to Non-Linear Filtering.” In Séminaire de Probabilités XXXIV. Lecture Notes in Mathematics 1729.

Doucet, Freitas, and Gordon. 2001. Sequential Monte Carlo Methods in Practice.

Doucet, Godsill, and Andrieu. 2000. “On Sequential Monte Carlo Sampling Methods for Bayesian Filtering.” Statistics and Computing.

E, Han, and Jentzen. 2017. “Deep Learning-Based Numerical Methods for High-Dimensional Parabolic Partial Differential Equations and Backward Stochastic Differential Equations.” Communications in Mathematics and Statistics.

Han, Jentzen, and E. 2018. “Solving High-Dimensional Partial Differential Equations Using Deep Learning.” Proceedings of the National Academy of Sciences.

Hartmann, Richter, Schütte, et al. 2017. “Variational Characterization of Free Energy: Theory and Algorithms.” Entropy.

Hutzenthaler, Jentzen, Kruse, et al. 2020. “Overcoming the Curse of Dimensionality in the Numerical Approximation of Semilinear Parabolic Partial Differential Equations.” Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

Hutzenthaler, and Kruse. 2020. “Multilevel Picard Approximations of High-Dimensional Semilinear Parabolic Differential Equations with Gradient-Dependent Nonlinearities.” SIAM Journal on Numerical Analysis.

Kebiri, Neureither, and Hartmann. 2019. “Adaptive Importance Sampling with Forward-Backward Stochastic Differential Equations.” In Stochastic Dynamics Out of Equilibrium. Springer Proceedings in Mathematics & Statistics.

Naesseth, Lindsten, and Schön. 2022. “Elements of Sequential Monte Carlo.” arXiv:1903.04797 [Cs, Stat].

Neal. 1998. “Annealed Importance Sampling.”

Nualart, and Schoutens. 2001. “Backward Stochastic Differential Equations and Feynman-Kac Formula for Lévy Processes, with Applications in Finance.” Bernoulli.

Nüsken, and Richter. 2021. “Solving High-Dimensional Hamilton–Jacobi–Bellman PDEs Using Neural Networks: Perspectives from the Theory of Controlled Diffusions and Measures on Path Space.” Partial Differential Equations and Applications.

Papanicolaou. 2019. “Introduction to Stochastic Differential Equations (SDEs) for Finance.” arXiv:1504.05309 [Math, q-Fin].

Pham. 2015. “Feynman-Kac Representation of Fully Nonlinear PDEs and Applications.” Acta Mathematica Vietnamica.

Zhao, Mair, Schön, et al. 2024. “On Feynman-Kac Training of Partial Bayesian Neural Networks.” In Proceedings of The 27th International Conference on Artificial Intelligence and Statistics.